Weighted Birkhoff averages: Deterministic and probabilistic perspectives

TL;DR

This work develops a unified, quantitative framework for weighted Birkhoff averages that accelerates time-averaging in quasi-periodic, almost periodic, and periodic systems via a compactly supported weighting $w_{p,q}$. It proves universal polynomial and exponential convergence rates (Theorem {['QMC']}) across finite- and infinite-dimensional tori for analytic observables, and introduces exponential convergence for computing Fourier coefficients (Theorem {['CLT-FOU']}) with an explicit notion of effective order. The probabilistic side is anchored by weighted laws of large numbers (Theorems {['LIMT1']} and {['LIMT2']}) and a weighted central limit theorem (Theorem {['CLTT1']}), built on classical results but adapted to the weighting framework. Collectively, these results bridge ergodic theory, harmonic analysis, and numerical methods, delivering rigorous, fast-converging tools for applications in celestial mechanics and dynamical systems, with practical implications for high-dimensional and infinite-dimensional settings.

Abstract

In this paper, we survey physically related applications of a class of weighted quasi-Monte Carlo methods from a theoretical, deterministic perspective, and establish quantitative universal rapid convergence results via various regularity assumptions. Specifically, we introduce weighting with compact support to the Birkhoff ergodic averages of quasi-periodic, almost periodic, and periodic systems, thereby achieving universal rapid convergence, including both arbitrary polynomial and exponential types. This is in stark contrast to the typically slow convergence in classical ergodic theory. As new contributions, we not only discuss more general weighting functions but also provide quantitative improvements to existing results; the explicit regularity settings facilitate the application of these methods to specific problems. We also revisit the physically related problems and, for the first time, establish universal exponential convergence results for the weighted computation of Fourier coefficients, in both finite-dimensional and infinite-dimensional cases. In addition to the above, we explore results from a probabilistic perspective, including the weighted strong law of large numbers and the weighted central limit theorem, by building upon the historical results.

Figures (5)

• Figure 1: The weighting procedure of the weighted quasi-Monte Carlo method for data generated from a quasi-periodic dynamical system
• Figure 2: The shape of the $1$-dimensional thickened torus $\mathbb{T}_\sigma ^1$ with $\sigma>0$
• Figure 3: The shape of the $j$-component space of the infinite-dimensional thickened torus $\mathbb{T}_\sigma ^\infty$ with $\sigma>0$
• Figure 4: Exponential convergence of the weighted quasi-Monte Carlo method in different settings. For the oscillating curves, red corresponds to $p = q = 1/2$, pink corresponds to $p = q = 1$, blue corresponds to $p = 1$ and $q = 2$, and green corresponds to $p = q = 2$, respectively.
• Figure 5: The geometric intuition of the conjugation \ref{['CONJU']} (with $d=1$) and the commutative diagram

Theorems & Definitions (6)

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